Fig. 5 Representation of the direction vector of a vanishing 
point and its corresponding lines in the image and 
in space 
There are numerous other possibilities to reason about 3D 
structures from image data and adequate hypothesis about the 
object’s form which may also be used (HARALICK 1989). 
The rules 1-6 are used to solve the first task of the geometric 
reasoning to determine the direction vectors of lines in space, 
while the rules 7-9 are used to derive the object’s shape. 
4 Rule-based System 
4.1 Concept 
The task of the vision system for interpreting images of 
polyhedra (ISIP) is to derive the object’s shape in space. 
A large variation of input data is admissable. There is no 
limitation in the complexitiy of the object as long as the planes 
of the object are connected and the object fullfills the 
assumption being approximable as a polyhedra. The system is 
able to evaluate perspective images as well as orthogonal ones. 
The exterior orientation of the image may vary, also the used 
camera, the input images may be terrestrical or aerial ones or 
images with short or long distances to the object. Because of 
this variety of admissable input data and the resulting big 
number, kind and order of procedures to be applied, there exist 
several strategies to solve the task. Therefore the vision system 
ISIP is organized in components of a rule-based system, 
finding one short way to the solution (NIEMANN 1981). 
The information, involved in the 2D sketch, and the 
information provided optionally by an operator is collected as 
the intermediate data of the system (Fig. 6). 
518 
The different rules within the interpretation system consist of a 
condition and a routine. The first conponent of the rule may 
either be a hypothesis about relationships or an assumption, 
resulting from the input data or the strategy of the 
interpretation process. The rules are based on apriori 
knowledge about the problem and their form of representation 
is known, so the operator could define knowledge in additional 
rules. The rules in the knowledge base are listed without a 
defined order, only the two components of each rule are 
connected together. The rules in the knowledge base of the 
system are devided into four classes. There are rules of the 
perspective geometry (cf. chap. 3.3) and rules, resulting from 
the strategy of the interpretation process or the used camera or 
the exterior orientation of the image. It will be shown how 
rules of the other three classes are applied to derive the object's 
shape (cf. chap. 4.4). 
The sequence of steps in which the relationships between 3D 
entities and the planes of the object are calculated, is 
determined by the control modul using the given 2D sketch 
and applying procedures of algebra and rules of the knowledge 
base. This central component of the system contains a special 
strategy and structure of order, handling the given knowledge, 
facts and rules, bottom up. The results of the system are 
declarations concerning the interpretation process and a 3D 
model of the object. 
4.2 Input of the system 
The data of the input images is stored in lists, according to the 
data structures of level 3 (cf. chap. 2). 
P:f(pnrx'y' ]..] 
L:{{inrbe}..} 
P is a list, consisting of n points with point number and 2-D 
coordinates (x’,y’). L is a list consisting of m lines. Each of 
these lines consists of a line number, the beginning and ending 
point of that line. 
E:{{lenrll...In}..} 
E contains the incidence relations used as hypothesis by the 
system. 
4.3 Control modul 
The system sucessively calculates the parameters of the planes 
of the object, stopping the process if all planes of the list E are 
determined in space respectively the points and the lines. The 
geometric reasoning depends on the geometrical constraints, 
which are either found automatically or given manually, on the 
actual status of knowledge, which is obtained so far, and on the 
parameters to be searched for. It is important to keep manual 
inputs by an operator as small as possible though in some cases 
inputs given by an operator may be necessary for the solution. 
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